If $\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1$, then the value of $x$ is

Given that $\sin\left(\sin^{-1} \frac{1}{5} + \cos^{-1} x\right) = 1$, the argument must satisfy $\sin^{-1} \frac{1}{5} + \cos^{-1} x = \frac{\pi}{2}$.

Using the identity $\sin^{-1} y + \cos^{-1} y = \frac{\pi}{2}$ for $y \in [-1, 1]$, we identify $x = \frac{1}{5}$.

Since $\frac{1}{5}$ lies within the domain of $\cos^{-1} x$, this solution is valid.